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Zbl 1026.82524
Chaves, A.S.
A fractional diffusion equation to describe Lévy flights.
(English)
[J] Phys. Lett., A 239, No.1-2, 13-16 (1998). ISSN 0375-9601

Summary: A fractional-derivatives diffusion equation is proposed that generates the Lévy statistics. The fractional derivatives are defined by the eigenvector equation $\partial_x^{\alpha}e^{ax}=a^{\alpha}e^{ax}$ and for one dimension the diffusion equation in an isotropic medium reads $$\partial_tn=(D/2)(\partial_x^{\alpha}+\partial_{-x}^{\alpha})n+v\partial_xn,\quad 1<\alpha\le 2.$$ The equation is based on a proposed generalization of Fick's law which reads $j=-(D/2)(\nabla_r^{\alpha-1}-\nabla_{-r}^{\alpha-1})n+vn$. The diffusion equation is also written for an anisotropic medium, and in this case it generates an asymmetric Lévy statistics.
MSC 2000:
*82C31 Stochastic methods in time-dependent statistical mechanics

Cited in: Zbl 1229.35313

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